Properties

Label 4-210e2-1.1-c1e2-0-8
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $2.81185$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s − 9-s + 4·11-s + 16-s + 12·19-s − 2·20-s − 25-s − 12·29-s − 4·31-s + 36-s + 4·41-s − 4·44-s − 2·45-s − 49-s + 8·55-s + 16·59-s − 20·61-s − 64-s − 12·71-s − 12·76-s + 24·79-s + 2·80-s + 81-s + 20·89-s + 24·95-s − 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 2.75·19-s − 0.447·20-s − 1/5·25-s − 2.22·29-s − 0.718·31-s + 1/6·36-s + 0.624·41-s − 0.603·44-s − 0.298·45-s − 1/7·49-s + 1.07·55-s + 2.08·59-s − 2.56·61-s − 1/8·64-s − 1.42·71-s − 1.37·76-s + 2.70·79-s + 0.223·80-s + 1/9·81-s + 2.11·89-s + 2.46·95-s − 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.81185\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486889448\)
\(L(\frac12)\) \(\approx\) \(1.486889448\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.67049834252611502316942207888, −12.04923802423061998963216648099, −11.50660462668572456523772294422, −11.46241751778221374578123629196, −10.57939948210151857012249908720, −10.09032348670532363079607383801, −9.421221370828641681268391980188, −9.197074951567093821059782080266, −9.118498699472213274919546382730, −8.085221601482870724138762015961, −7.38038611945640380530025546703, −7.31401588725773932519732375859, −6.16253960391928455211864751269, −5.93081486344308861734746911120, −5.28833018134119284187123702292, −4.79178130173635207949765314522, −3.60825502339670272635639914580, −3.50514779740060973345442696270, −2.19614907503391920152352820609, −1.23492869683336486031404324122, 1.23492869683336486031404324122, 2.19614907503391920152352820609, 3.50514779740060973345442696270, 3.60825502339670272635639914580, 4.79178130173635207949765314522, 5.28833018134119284187123702292, 5.93081486344308861734746911120, 6.16253960391928455211864751269, 7.31401588725773932519732375859, 7.38038611945640380530025546703, 8.085221601482870724138762015961, 9.118498699472213274919546382730, 9.197074951567093821059782080266, 9.421221370828641681268391980188, 10.09032348670532363079607383801, 10.57939948210151857012249908720, 11.46241751778221374578123629196, 11.50660462668572456523772294422, 12.04923802423061998963216648099, 12.67049834252611502316942207888

Graph of the $Z$-function along the critical line